Factor the following expression: $-6$ $x^2$ $-11$ $x+$ $21$
Solution: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-6)}{(21)} &=& -126 \\ {a} + {b} &=& & & {-11} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-126$ and add them together. Remember, since $-126$ is negative, one of the factors must be negative. The factors that add up to ${-11}$ will be your ${a}$ and ${b}$ When ${a}$ is ${7}$ and ${b}$ is ${-18}$ $ \begin{eqnarray} {ab} &=& ({7})({-18}) &=& -126 \\ {a} + {b} &=& {7} + {-18} &=& -11 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-6}x^2 +{7}x {-18}x +{21} $ Group the terms so that there is a common factor in each group: $ ({-6}x^2 +{7}x) + ({-18}x +{21}) $ Factor out the common factors: $ x(-6x + 7) + 3(-6x + 7) $ Notice how $(-6x + 7)$ has become a common factor. Factor this out to find the answer. $(-6x + 7)(x + 3)$